A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications

The main contribution of this paper is to present a new sufficient condition for the subexponential asymptotics of the stationary distribution of a GI/GI/1-type Markov chain without jumps from level "infinity" to level zero. For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains without disasters} because they are often used to analyze semi-Markovian queues without "disasters", which are negative customers who remove all the customers in the system (including themselves) on their arrivals. In this paper, we demonstrate the application of our main result to the stationary queue length distribution in the standard BMAP/GI/1 queue. Thus we obtain new asymptotic formulas and prove the existing formulas under weaker conditions than those in the literature. In addition, applying our main result to a single-server queue with Markovian arrivals and the $(a,b)$-bulk-service rule (i.e., MAP/${\rm GI}^{(a,b)}$/1 queue), we obatin a subexponential asymptotic formula for the stationary queue length distribution.Comment: Submitted for revie

Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments

This paper considers the tail asymptotics for a cumulative process $\{B(t); t \ge 0\}$ sampled at a heavy-tailed random time $T$. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality ${\sf P}(B(T) > bx) \sim {\sf P}(M(T) > bx) \sim {\sf P}(T>x)$ as $x \to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certain positive constant. The main results of this paper can be used to obtain the subexponential asymptotics for various queueing models in Markovian environments. As an example, using the main results, we derive subexponential asymptotic formulas for the loss probability of a single-server finite-buffer queue with an on/off arrival process in a Markovian environment

Importance sampling of heavy-tailed iterated random functions

We consider a stochastic recurrence equation of the form $Z_{n+1} = A_{n+1} Z_n+B_{n+1}$, where $\mathbb{E}[\log A_1]<0$, $\mathbb{E}[\log^+ B_1]<\infty$ and $\{(A_n,B_n)\}_{n\in\mathbb{N}}$ is an i.i.d. sequence of positive random vectors. The stationary distribution of this Markov chain can be represented as the distribution of the random variable $Z \triangleq \sum_{n=0}^\infty B_{n+1}\prod_{k=1}^nA_k$. Such random variables can be found in the analysis of probabilistic algorithms or financial mathematics, where $Z$ would be called a stochastic perpetuity. If one interprets $-\log A_n$ as the interest rate at time $n$, then $Z$ is the present value of a bond that generates $B_n$ unit of money at each time point $n$. We are interested in estimating the probability of the rare event $\{Z>x\}$, when $x$ is large; we provide a consistent simulation estimator using state-dependent importance sampling for the case, where $\log A_1$ is heavy-tailed and the so-called Cram\'{e}r condition is not satisfied. Our algorithm leads to an estimator for $P(Z>x)$. We show that under natural conditions, our estimator is strongly efficient. Furthermore, we extend our method to the case, where $\{Z_n\}_{n\in\mathbb{N}}$ is defined via the recursive formula $Z_{n+1}=\Psi_{n+1}(Z_n)$ and $\{\Psi_n\}_{n\in\mathbb{N}}$ is a sequence of i.i.d. random Lipschitz functions

Asymptotic behavior of the loss probability for an M/G/1/N queue with vacations

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N -(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model